The positive definite geometric programming is transferred into a nolinear programming with constraints of linear equality by duality principle.

Self-Insurance,Self-Protection and Market Insurance within the Dual Theory of Choice;

According to the dual theory,a simple geometric programming was proposed to derive a corresponding geometric dual problem instead of cross-entropy optimization problem with cross-entropy constraints,which is a concave programming one with linear constrains,leading to a simpler calculation.

In this paper, a new Hopfields-type neural network for linear programming, which is based on dual theory and makes use of Sigmoid function, is presented.

An Application of Duality Theory to The Analysis of Economic Equlilbrium;

This paper discuses the relationship between the saddle point of Lagrange function and the optimization solving of the primal optimization problem and its dual problem,with verification of some new features in duality theory.

Discuss the relations between the quadratic semi-definite programming and the semi-definite least squares problem,and study the primal-dual interior point algorithm for this programing based on the duality theory,and give the proof of the unique solution based on the NT search direction.

Furthermore the sufficient optimality conditions and duality results are obtained for a nonlinear programming problem involving B-preinvex functions.

In this paper,we extend the restricted set X of the programming from open set to locally starshaped set,and extend the η convexity to the η convexity for non differentiable functions,and research on the optimality necessary conditions and the duality of the multi objective programming.

From semidefinite programming duality theory, conditions for infeasibility of the linear matrix inequalitys as well as dual opti.

Pian Ou theory is the important part of ancient literature theoy.